Let’s be honest: if not for our parents, none of us would be here. Therefore, we have to respect our parents for the very fact that we exist at all. Not to do so constitutes the ultimate ingratitude. Extrapolating this one step further, neither we nor our parents (nor their parents, etc.) would be here if not for God. By honoring our parents, we are in part honoring Him.
The Ten Commandments were written with five commandments on each of the two tablets. The first tablet contained laws applying to a person with respect to God, such as to observe Shabbos and not to serve idols. The other tablet contained laws between one person and another, such as not to kill or to steal. The obligation to honor our parents is on the first tablet because it truly is a matter between man and God, not just between us and our folks.
An obligation to “honor” someone is a little vague. Are we supposed to throw a testimonial dinner for our parents, or perhaps erect a statue of them? Happily, the Talmud defines the parameters of “honor” for us: one must see that his parents have food, clothes and shelter, plus attend to them and escort them (Kiddushin 31b). One may spend the parents’ money on these needs, but if the parents do not have the means, a person must support them himself. If one sees a parent transgressing a Torah law, he must correct his parent in a gentle manner (for example, by asking, “Father, doesn’t the Torah say…?).
This mitzvah applies to both men and women, in all times and places, although there is a limitation when it comes to women: after she is married, her ability to do so may conflict with her duties to her own husband and family. (Her husband would therefore be responsible to ensure that his in-laws’ honor is met – see Kitzur Shulchan Aruch 143:20.)
The obligation to honor one’s parents is discussed in the Talmud in the tractate of Kiddushin, 30b-32a. In the Shulchan Aruch, it can be found in Yoreh Deah 240. It is #210 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos and #41 of the 77 positive mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.